G. D. SMITH Brunei University Numerical Solution of partial differential equations FINITE DIFFERENCE METHODS THIRD EDITION CLARENDON PRESS OXFORD
Contents NOTATION 1. INTRODUCTION AND FINITE-DIFFERENCE FORMULAE Descriptive treatment of elliptic equations 1 Descriptive treatment of parabolic and hyperbolic equations 4 Finite-difference approximations to derivatives 6 Notation for functions of several variables 8 2. PARABOLIC EQUATIONS: FINITE DIFFERENCE METHODS, CONVERGENCE, AND STABILITY Transformation to non-dimensional form 11 An explicit finite-difference approximation to SU/dt = d 2 U/dx 2 12 A worked example covering three cases and including comparison tables 13 Crank-Nicolson implicit method 19 Worked example including a comparison table 21 Solution of the implicit equations by Gauss's elimination method 24 The stability of the elimination method 27 A weighted average approximation 28 Derivative boundary conditions 29 Worked examples including comparison tables: (i) Explicit formula and central-differenced boundary condition 31 (ii) Explicit formula and forward-differenced boundary condition 33 (iii) Implicit formula and central-differenced boundary condition 36 The local truncation error and a worked example 38 Consistency and a worked example illustrating both consistency and inconsistency 40 Convergence, descriptive treatment and the analysis of an explicit approximation 43 Stability, descriptive treatment 47
Vlll Contents Vector and matrix norms, subordinate matrix norms, p(a)=s l A 49 A necessary and sufficient condition for stability, A =sl, and two worked examples 51 Matrix method of analysis, fixed mesh size. 57 A note on the eigenvalues of /(A) and [/i(a)] _1 /2(A) 58 The eigenvalues of a common tridiagonal matrix 59 Theorems on bounds for eigenvalues and an application. (Gerschgorin's theorems) 60 Gerschgorin's circle theorem and the norm of matrix A 62 Stability criteria for derivative boundary conditions using (i) the circle theorem (ii) A»«1 63 Stability condition allowing exponential growth 66 Stability, von Neumann's method, and three worked examples 67 The global rounding error 71 Lax's equivalence theorem (statement only) and a detailed analysis of a simple case 72 Finite-difference approximations to du/dt = V 2 U in cylindrical and spherical polar co-ordinates 75 A worked example involving lim(3fj/öx)/x 77 Exercises and Solutions 79 PARABOLIC EQUATIONS: ALTERNATIVE DERIVATION OF DIFFERENCE EQUATIONS AND MISCELLANEOUS TOPICS Reduction to a System of ordinary differential equations 111 A note on the Solution of dv/dt = AV + b 113 Finite-difference approximations via the ordinary differential equations 115 The Pade approximants to exp 0 116 Standard finite-difference equations via the Pade approximants 117 A 0 -stability, L 0 -stability and the symbol of the method 119 A necessary constraint on the time step for the Crank- Nicolson method 122 The local truncation errors associated with the Pade approximants 124 Stiff equations 126 An extrapolation method for improving accuracy in t 126 The symbol for the extrapolation method 128 The arithmetic of the extrapolation method 129 The local truncation errors and Symbols of extrapolation schemes 132
Contents The eigenvalue-eigenvector Solution of a System of ordinary differential equations (i) Preliminary results 132 (ii) The eigenvalue-eigenvector Solution of dv/df = AV 134 (iii) An application giving an approximate Solution for large t 135 Miscellaneous methods for improving accuracy: (i) Reduction of the local truncation error - the Douglas equations 137 (ii) Use of three time-level difference equations 138 (iii) Deferred correction method 139 (iv) Richardson's deferred approach to the limit 141 Solution of non-linear parabolic equations: (i) Newton's linearization method and a worked example 142 (ii) Richtmyer's linearization method 144 (iii) Lee's three time-level method 146 A comparison of results for methods (i), (ii), and (iii) for a particular problem 147 The stability of three or more time-level difference equations: (i) A useful theorem on eigenvalues 148 (ii) Matrices with common eigenvector Systems 150 (iii) A worked example 150 Introduction to the analytical Solution of homogeneous difference equations: 153 (i) The eigenvalues and vectors of a common tridiagonal matrix 154 (ii) The analytical Solution of the classical explicit approximation to du/dt = d 2 U/dx 2 156 Exercises and Solutions 158 ix HYPERBOLIC EQUATIONS AND CHARACTERISTICS Analytical Solution of first-order quasi-linear equations 175 A worked example and discussion 176 Numerical Integration along a characteristic 178 A worked example 179 Finite-difference methods on a rectangular mesh for firstorder equations: (i) Lax-Wendroff explicit method and a worked example with a comparison table 181 (ii) Lax-Wendroff method for a set of simultaneous equations 183 (iii) The Courant-Friedrichs-Lewy condition 186
X Contents (iv) Wendroff's implicit approximation 187 Propagation of discontinuities, first-order equations: (i) Discontinuous initial values 188 (ii) Discontinuous initial derivatives 189 Discontinuities and finite-difference approximations. An example using Wendroff's implicit approximation 190 Reduction of a first-order equation to a System of ordinary differential equations 193 The (1,0) Pade difference approximation 195 A comment on the non-stiffness of the equations 196 The (1, 1) Pade or Crank-Nicolson difference equations 196 An improved approximation to du/dx and the (1,0) Pade difference equations 197 A word of caution on the central-difference approximation to du/dx 200 Second-order quasi-linear hyperbolic equations. Characteristic curves, and the differential relationship along them 202 Numerical Solution by the method of characteristics 204 A worked example 207 A characteristic as an initial curve 209 Propagation of discontinuities, second-order equations 210 Finite-difference methods on a rectangular mesh for secondorder equations: 213 (i) Explicit methods and the Courant-Friedrichs-Lewy condition 213 (ii) Implicit methods with particular reference to the waveequation 216 Simultaneous first-order equations and stability 217 Exercises and Solutions 220 5. ELLIPTIC EQUATIONS AND SYSTEMATIC ITERATIVE METHODS Introduction 239 Worked examples: (i) A torsion problem. (ii) A heatconduction problem with derivative boundary conditions 240 Finite-differences in polar co-ordinates 245 Formulae for derivatives near a curved boundary Improvement of the accuracy of Solutions: (i) Finer mesh. (ii) Deferred approach to the limit. (iii) Deferred correction method. (iv) More accurate finite-difference formulae including the nine-point formula 248 Analysis of the discretization error of the five-point approximation to Poisson's equation over a rectangle. Quoted 252 result for irregulär boundaries 254
Contents XI Comments on the Solution of difference equations, covering Gauss elimination, LU decomposition, rounding errors, illconditioning, iterative refinement, iterative methods 257 Systematic iterative methods for large linear Systems 260 Jacobi, Gauss-Seidel, and SOR methods 261 A worked example covering each method 263 Jacobi, Gauss-Seidel, and SOR methods in matrix form 266 A necessary and sufficient condition for convergence of iterative methods 268 A sufficient condition for convergence 269 Asymptotic and average rates of convergence 270 Methods for accelerating convergence. (i) Lyusternik's method. (ii) Aitken's method. An illustrative example 272 Eigenvalues of the Jacobi and SOR iteration matrices and two worked examples 275 The Optimum acceleration parameter for the SOR method. A necessary theorem 277 Proof of (A. + cö 1) 2 = Ao> 2 /j, 2 for block tridiagonal coefficient matrices 279 Non-zero eigenvalues of the Jacobi iteration matrix 280 Theoretical determination of the optimum relaxation parameter w b 282 Calculation of (o b for a rectangle and other Solution domains 285 The Gauss-Seidel iteration matrix H(l) 285 Re-ordering of equations and unknowns 286 Point iterative methods and re-orderings 287 Introduction to 2-cyclic matrices and consistent ordering 288 2-cyclic matrices 289 Ordering vectors for 2-cyclic matrices 290 Consistent ordering of a 2-cyclic matrix 292 The ordering vector for a block tridiagonal matrix 294 An example of a consistently ordered 2-cyclic matrix that is not block tridiagonal 297 Additional comments on consistent ordering and the SOR method 297 Consistent orderings associated with the five-point approximation to Poisson's equation 298 Stone's strongly implicit iterative method 302 A recent direct method 309 Exercises and Solutions 311 INDEX 334